A013929 -id:A013929 - cp's OEIS frontend

A376590 Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 3, -2, 0, 0, -1, 0, 1, -1, 2, -2, 0, 1, -1, 0, 1, -1, 2, -2, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 1, 2, -3, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 2, -2, 3, -2, -1

Offset: 1

Gus Wiseman, Oct 01 2024

sign

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...

The version for A000002 is A376604, first differences of A054354.
The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
Zeros are A376591, complement A376592.
Sorted positions of first appearances are A376655.
A000040 lists the prime numbers, differences A001223.
A001597 lists perfect-powers, complement A007916.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For squarefree numbers: A076259 (first differences), A376591 (inflections and undulations), A376592 (nonzero curvature), A376655 (sorted first positions).
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

Differences[Select[Range[100],SquareFreeQ],2]

from math import isqrt
from sympy import mobius
def A376590(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    a = iterfun(f,n)
    b = iterfun(lambda x:f(x)+1,a)
    return a+iterfun(lambda x:f(x)+2,b)-(b<<1) # Chai Wah Wu, Oct 02 2024

A007675 Numbers m such that m, m+1 and m+2 are squarefree.

Original entry on oeis.org

1, 5, 13, 21, 29, 33, 37, 41, 57, 65, 69, 77, 85, 93, 101, 105, 109, 113, 129, 137, 141, 157, 165, 177, 181, 185, 193, 201, 209, 213, 217, 221, 229, 237, 253, 257, 265, 281, 285, 301, 309, 317, 321, 329, 345, 353, 357, 365, 381, 389, 393, 397, 401, 409, 417, 429, 433, 437, 445, 453

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Four categories: all terms are composites like {33, 34, 35}; first term only is prime like {37, 38, 39}; third term only is prime like {57, 58, 59}; first and third are primes like {29, 30, 31}. - Labos Elemer
Four consecutive integers cannot be squarefree as one of them is divisible by 2^2 = 4. - Amarnath Murthy, Feb 18 2002
Numbers m such that m^3 + 3m^2 + 2m is squarefree. See proof below. - Charles R Greathouse IV, Mar 05 2013
There are kx + O(x/log x) terms of this sequence below x, where k = A206256. - Charles R Greathouse IV, Mar 05 2013
Proof: m^3 + 3m^2 + 2m = m*(m+1)*(m+2) and the factors are pairwise relatively prime, because (m+1) is even. - Thomas Ordowski, Apr 20 2013
Conjecture: for every prime p, the numbers p# - 1, p#, p# + 1 are squarefree, where primorial p# = product of all primes <= p. - Thomas Ordowski, Apr 21 2013
Let f(m) = abs(mu(m*(m+1)*(m+2))), where mu(m) is the Moebius function, then the sum S(m) = f(1) + f(2) + ... + f(m) ~ k*m with the constant k = A206256 = 0.12548698.... - Thomas Ordowski, Apr 22 2013
All terms are congruent to 1 (mod 4). - Zak Seidov, Dec 22 2014

			85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.

P. R. Halmos, Problems for Mathematicians Young and Old. Math. Assoc. America, 1991, p. 28.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Subsequence of A007674, A016813, and A056911.

Cf. A005117, A013929, A008966, A007531.

a007675 n = a007675_list !! (n-1)
a007675_list = f 1 a008966_list where
   f n (u:xs'@(v:w:x:xs)) | u == 1 && w == 1 && v == 1 = n : f (n+4) xs
                          | otherwise = f (n+1) xs'
-- Reinhard Zumkeller, Nov 05 2011

select(t -> andmap(NumberTheory:-IsSquareFree,[t,t+1,t+2]), [seq(i,i=1..1000,4)]); # Robert Israel, Jul 16 2024

Select[Range[1000], SquareFreeQ[#(# + 1)(# + 2)] &] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
Transpose[Select[Partition[Select[Range[400], SquareFreeQ], 3, 1], Differences[#] == {1, 1} &]][[1]] (* Harvey P. Dale, Apr 11 2012 *)
Select[Range[1, 499, 2], MoebiusMu[#^3 + 3#^2 + 2#] != 0 &] (* Alonso del Arte, Jan 16 2014 *)
SequencePosition[Table[If[SquareFreeQ[n],1,0],{n,500}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 14 2017 *)

is(n)=issquarefree(n)&&issquarefree(n+1)&&issquarefree(n+2) \\ Charles R Greathouse IV, Mar 05 2013

Numbers m such that g(m)*g(m+1)*g(m+2) = 1, where g(w) = abs(mu(w)). - Labos Elemer

a(n) ~ c*n with c = 7.96895... = 1/A206256. - Charles R Greathouse IV, Mar 05 2013

A091050 Number of divisors of n that are perfect powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1

Offset: 1

Reinhard Zumkeller, Dec 15 2003

nonn

Not the same as A005361: a(72)=5 <> A005361(72)=6.

			Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108},
a(108) = #{1^2, 2^2, 3^2, 3^3, 6^2} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A000005, A001597, A005361, A072102, A091051.

a091050 = sum . map a075802 . a027750_row
-- Reinhard Zumkeller, Dec 13 2012

ppQ[n_] := GCD @@ Last /@ FactorInteger@ n > 1; ppQ[1] = True; f[n_] := Length@ Select[ Divisors@ n, ppQ]; Array[f, 105] (* Robert G. Wilson v, Dec 12 2012 *)

a(n) = 1+ sumdiv(n, d, ispower(d)>1); \\ Michel Marcus, Sep 21 2014

a(n)={my(f=factor(n)[,2]); 1 + if(#f, sum(k=2, vecmax(f), moebius(k)*(1 - prod(i=1, #f, 1 + f[i]\k))))} \\ Andrew Howroyd, Aug 30 2020

a(n) = 1 iff n is squarefree: a(A005117(n)) = 1, a(A013929(n)) > 1.
a(p^k) = k for p prime, k>0: a(A000961(n)) = A025474(n).
a(n) = Sum_{k=1..A000005(n)} A075802(A027750(n,k)). - Reinhard Zumkeller, Dec 13 2012
G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 20 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + A072102 = 1.874464... . - Amiram Eldar, Dec 31 2023

Wrong formula deleted by Amiram Eldar, Apr 29 2020

A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).

Original entry on oeis.org

1, 1, 4, 4, 9, 12, 16, 18, 20, 28, 28, 36, 40, 40, 45, 52, 56, 60, 64, 68, 72, 76, 81, 88, 96, 100, 100, 104, 108, 112, 126, 128, 136, 136, 148, 150, 156, 162, 164, 172, 176, 180, 189, 192, 196, 198, 208, 220, 225, 228, 232, 236, 240, 250, 256, 261, 268, 270

Offset: 1

Gus Wiseman, Nov 16 2024

nonn

			The terms together with their prime indices begin:
    1: {}
    1: {}
    4: {1,1}
    4: {1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   28: {1,1,4}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   40: {1,1,1,3}
   45: {2,2,3}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   68: {1,1,7}
   72: {1,1,1,2,2}

Terms appearing twice are A061351 + 1.
For prime-powers we have A065514 (diffs A377781), opposite A345531 (diffs A377703).
For squarefree we have A112925 (differences A378038).
The opposite for squarefree is A112926 (differences A378037).
The opposite is A377783 (union A378040), restriction of A120327 (differences A378039).
Restriction of A378033, which has differences A378036.
The first-differences are A378034, opposite A377784.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061398 counts squarefree numbers between primes (sums A337030), zeros A068360.
A061399 counts nonsquarefree numbers between primes (sums A378086), zeros A068361.
A070321 gives the greatest squarefree number up to n.
A377046 encodes k-differences of nonsquarefree numbers, zeros A377050.
Cf. A053797, A053806, A072284, A065890, A224363, A366833, A377431.
Cf. A377047, A377048, A377049, A377430, A378082, A378083, A378084.

Table[NestWhile[#-1&,Prime[n],#>1&&SquareFreeQ[#]&],{n,100}]

a(n) = A378033(prime(n)).

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

Original entry on oeis.org

3, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2

Offset: 1

Labos Elemer, Oct 29 2001

nonn

Seems identical to A054546. Each odd prime arises once or twice!?

First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - Gus Wiseman, Sep 15 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A054576, A065308-A065309.
For twin 2's see A169643.
Positions of 1's are A375926, complement A014689 (except first term).
Other families of numbers and their first-differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310 (this).
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A046933, A053707, A054546, A110969, A176246, A251092, A373403, A375929.

t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]
Differences[Select[Range[100],!PrimeQ[#]&]] (* Gus Wiseman, Sep 15 2024 *)

{ p=1; f=2; m=1; for (n=1, 1000, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); write("b065310.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 16 2009

A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

Original entry on oeis.org

4, 8, 4, 9, 1, -3, 12, 3, 2, 5, 16, 4, 1, -1, -6, 18, 2, -2, -3, -2, 4, 20, 2, 0, 2, 5, 7, 3, 24, 4, 2, 2, 0, -5, -12, -15, 25, 1, -3, -5, -7, -7, -2, 10, 25, 27, 2, 1, 4, 9, 16, 23, 25, 15, -10, 28, 1, -1, -2, -6, -15, -31, -54, -79, -94, -84, 32, 4, 3, 4, 6, 12, 27, 58, 112, 191, 285, 369

Offset: 0

Gus Wiseman, Oct 19 2024

Row k is the k-th differences of A013929.

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ---------------------------------------------------------
  k=0:   4     8     9    12    16    18    20    24    25
  k=1:   4     1     3     4     2     2     4     1     2
  k=2:  -3     2     1    -2     0     2    -3     1    -1
  k=3:   5    -1    -3     2     2    -5     4    -2     4
  k=4:  -6    -2     5     0    -7     9    -6     6    -7
  k=5:   4     7    -5    -7    16   -15    12   -13    10
  k=6:   3   -12    -2    23   -31    27   -25    23   -13
  k=7: -15    10    25   -54    58   -52    48   -36    13
  k=8:  25    15   -79   112  -110   100   -84    49     1
  k=9: -10   -94   191  -222   210  -184   133   -48   -57
Triangle form:
   4
   8   4
   9   1  -3
  12   3   2   5
  16   4   1  -1  -6
  18   2  -2  -3  -2   4
  20   2   0   2   5   7   3
  24   4   2   2   0  -5 -12 -15
  25   1  -3  -5  -7  -7  -2  10  25
  27   2   1   4   9  16  23  25  15 -10
  28   1  -1  -2  -6 -15 -31 -54 -79 -94 -84
  32   4   3   4   6  12  27  58 112 191 285 369

Initial rows: A013929, A078147, A376593.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, sums A377039, absolute A377040.
Triangle row-sums are A377047, absolute version A377048.
Column n = 1 is A377049, for squarefree A377041, for prime A007442 or A030016.
First position of 0 in each row is A377050.
For prime-power instead of nonsquarefree we have A377051.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A120992, A376311, A376591, A376592.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A013929(i+k).

A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 20, 24, 27, 29, 31, 33, 37, 38, 42, 45, 46, 50, 52, 56, 61, 62, 64, 67, 68, 71, 78, 81, 84, 86, 92, 93, 96, 100, 103, 105, 109, 110, 117, 118, 121, 122, 130, 139, 141, 142, 145, 149, 150, 154, 158, 162, 166, 167, 170, 172, 174, 180

Offset: 1

Labos Elemer, May 24 2002

nonn

Also the number of squarefree numbers <= prime(n). - Gus Wiseman, Dec 08 2024

			a(25)=61 because A005117(61) = prime(25) = 97.
From _Gus Wiseman_, Dec 08 2024: (Start)
The squarefree numbers up to prime(n) begin:
n = 1  2  3  4   5   6   7   8   9  10
    ----------------------------------
    2  3  5  7  11  13  17  19  23  29
    1  2  3  6  10  11  15  17  22  26
       1  2  5   7  10  14  15  21  23
          1  3   6   7  13  14  19  22
             2   5   6  11  13  17  21
             1   3   5  10  11  15  19
                 2   3   7  10  14  17
                 1   2   6   7  13  15
                     1   5   6  11  14
                         3   5  10  13
                         2   3   7  11
                         1   2   6  10
                             1   5   7
                                 3   6
                                 2   5
                                 1   3
                                     2
                                     1
The column-lengths are a(n).
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A013928.
The strict version is A112929.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A070321 gives the greatest squarefree number up to n.
Other families: A014689, A027883, A378615, A065890.
Squarefree numbers between primes: A061398, A068360, A373197, A373198, A377430, A112925, A112926.
Nonsquarefree numbers: A057627, A378086, A061399, A068361, A120327, A377783, A378032, A378033.
Cf. A046933, A049093, A053797, A072284, A077641, A224363, A337030, A345531, A008864.

Position[Select[Range[300], SquareFreeQ], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Aug 17 2023 *)

lista(nn)=sqfs = select(n->issquarefree(n), vector(nn, i, i)); for (i = 1, #sqfs, if (isprime(sqfs[i]), print1(i, ", "));); \\ Michel Marcus, Sep 11 2013

a(n,p=prime(n))=sum(k=1, sqrtint(p), p\k^2*moebius(k)) \\ Charles R Greathouse IV, Sep 13 2013

a(n,p=prime(n))=my(s); forfactored(k=1, sqrtint(p), s+=p\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Nov 27 2017

first(n)=my(v=vector(n),pr,k); forsquarefree(m=1,n*logint(n,2)+3, k++; if(m[2][,2]==[1]~, v[pr++]=k; if(pr==n, return(v)))) \\ Charles R Greathouse IV, Jan 08 2018

from math import isqrt
from sympy import prime, mobius
def A071403(n): return (p:=prime(n))+sum(mobius(k)*(p//k**2) for k in range(2,isqrt(p)+1)) # Chai Wah Wu, Jul 20 2024

A005117(a(n)) = A000040(n) = prime(n).
a(n) ~ (6/Pi^2) * n log n. - Charles R Greathouse IV, Nov 27 2017
a(n) = A013928(A008864(n)). - Ridouane Oudra, Oct 15 2019
From Gus Wiseman, Dec 08 2024: (Start)
a(n) = A112929(n) + 1.
a(n+1) - a(n) = A373198(n) = A061398(n) - 1.
(End)

A190641 Numbers having exactly one non-unitary prime factor.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Reinhard Zumkeller, Dec 29 2012

nonn

Numbers k such that the powerful part of k, A057521(k), is a composite prime power (A246547). - Amiram Eldar, Aug 01 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio.
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, Iss. 1 (2011), pp. 52-66. See p. 61.

Subsequence of A013929 and of A327877.

Cf. A056170, A057521, A154945, A246547, A359466 (characteristic function).

a190641 n = a190641_list !! (n-1)
a190641_list = map (+ 1) $ elemIndices 1 a056170_list

Select[Range[164],Count[FactorInteger[#][[All, 2]], 1] == Length[FactorInteger[#]] - 1 &] (* Geoffrey Critzer, Feb 05 2015 *)

list(lim)=my(s=lim\4, v=List(), u=vectorsmall(s, i, 1), t, x); forprime(k=2, sqrtint(s), t=k^2; forstep(i=t, s, t, u[i]=0)); forprime(k=2, sqrtint(lim\1), for(e=2,logint(lim\1,k), t=k^e; for(i=1, #u, if(u[i] && gcd(k, i)==1, x=t*i; if(x>lim, break); listput(v, x))))); Set(v) \\ Charles R Greathouse IV, Aug 02 2016

isok(n) = my(f=factor(n)); #select(x->(x>1), f[,2]) == 1; \\ Michel Marcus, Jul 30 2017

A056170(a(n)) = 1.

a(n) ~ k*n, where k = Pi^2/(6*A154945) = 2.9816096.... - Charles R Greathouse IV, Aug 02 2016

A373127 Length of the n-th maximal antirun of squarefree numbers differing by more than one.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 5, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 1, 2, 1, 2, 4, 2, 1, 4, 1, 3, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 3, 4, 1, 2, 1

Offset: 1

Gus Wiseman, Jun 05 2024

nonn

The sum of this antirun is given by A373411.

An antirun of a sequence (in this case A005117) is an interval of positions at which consecutive terms differ by more than one.

			Row-lengths of:
   1
   2
   3  5
   6
   7 10
  11 13
  14
  15 17 19 21
  22
  23 26 29
  30
  31 33
  34
  35 37
  38
  39 41
  42
  43 46
  47 51 53 55 57

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

Positions of first appearances are A373128, sorted A373200.
Functional neighbors: A007674, A027833 (partial sums A029707), A120992, A373403, A373408, A373409, A373411.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A077643 counts squarefree numbers with n bits, sum A373123.
Cf. A049093, A049094, A061398, A077641, A112925, A112926, A174965, A350842, A372683, A373125, A373126, A373197, A373198.

Length/@Split[Select[Range[100],SquareFreeQ],#1+1!=#2&]

A008477 If n = Product (p_j^k_j) then a(n) = Product (k_j^p_j).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 8, 1, 1, 4, 1, 1, 1, 16, 1, 8, 1, 4, 1, 1, 1, 9, 32, 1, 27, 4, 1, 1, 1, 25, 1, 1, 1, 32, 1, 1, 1, 9, 1, 1, 1, 4, 8, 1, 1, 16, 128, 32, 1, 4, 1, 27, 1, 9, 1, 1, 1, 4, 1, 1, 8, 36, 1, 1, 1, 4, 1, 1, 1, 72, 1

Offset: 1

Olivier Gérard

For any n, the sequence n, a(n), a(a(n)), a(a(a(n))), ... is eventually periodic with period <= 2 [Farrokhi]. - N. J. A. Sloane, Apr 25 2009
a(A005117(n)) = 1; a(A013929(n)) > 1; A010052(a(A122132(n))) = 1. - Reinhard Zumkeller, Feb 17 2012
From Bernard Schott, Mar 26 2021: (Start)
The study of some properties of this sequence was proposed in the 1st problem of Concours Général in 2012 in France (see links).
Terms are precisely the powerful numbers in A001694.
If m is a term, there is a term q such that a(q) = m.
a(a(n)) <= n (see examples). (End)

			For n = 24 = 2^3*3^1, a(24) = 3^2*1^3 = 9, so a(9) = 2^3 = 8 and a(a(24)) = 8 < 24.
For n = 243 = 3^5, a(243) = 5^3 = 125, so a(125) = 3^5 = 243 and a(a(243)) = 243.

M. F. Hasler, Table of n, a(n) for n = 1..10000
Annales Concours Général, Sujet Concours Général 2012 (in French, problems).
Annales Concours Général, Corrigé Concours Général 2012 (in French, solutions).
M. Farrokhi, The Prime Exponentiation of an Integer: Problem 11315, Amer. Math. Monthly, 116 (2009), 470. - from _N. J. A. Sloane_, Apr 25 2009
Index to sequences related to Olympiads and other Mathematical Competitions.

Cf. A001694, A027748, A124010.
Cf. A005117, A013929, A010052, A062307, A122132, A342551.
Cf. A008478 (fixed points).

⍝ Dyalog dialect
A008477 ← {×/{⍺*⍨≢⍵}⌸factors(⍵)} ⍝ Needs also factors function from https://dfns.dyalog.com/c_factors.htm - Antti Karttunen, Feb 16 2024

a008477 n = product $ zipWith (^) (a124010_row n) (a027748_row n)
-- Reinhard Zumkeller, Feb 17 2012

A008477 := proc(n) local e,j; e := ifactors(n)[2]:
mul (e[j][2]^e[j][1], j=1..nops(e)) end:
seq (A008477(n), n=1..60);
# Peter Luschny, Jan 17 2010

Prepend[ Array[ Times @@ Map[ Power @@ RotateLeft[ #1, 1 ]&, FactorInteger[ # ] ]&, 100, 2 ], 1 ]
Table[Times@@(First[#]^Last[#]&/@Transpose[Reverse[ Transpose[ FactorInteger[ n]]]]),{n,80}] (* Harvey P. Dale, Jul 22 2014 *)

A008477(n)=factorback(factor(n)*[0,1;1,0]) \\ M. F. Hasler, May 20 2012

from sympy import factorint, prod
a = lambda n: prod([pk[1]**pk[0] for pk in factorint(n).items()])
print([a(n) for n in range(1,61)]) # Darío Clavijo, Nov 06 2023

Multiplicative with a(p^e) = e^p. - David W. Wilson, Aug 01 2001

cp's OEIS Frontend

A376590 Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

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A007675 Numbers m such that m, m+1 and m+2 are squarefree.

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A091050 Number of divisors of n that are perfect powers.

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A378032 a(1) = a(2) = 1; a(n>2) is the greatest nonsquarefree number < prime(n).

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A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

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A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

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A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

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